Speaker: Connor Meehan
Title: Infinite Games and Analytic Sets
Abstract: In the context of set theory, infinite games have been studied since the mid-20th century and have created an interesting web of connections, such as with measurable cardinals. Upon specifying a subset A of sequences of natural numbers, an infinite game G(A) involves two players alternately choosing natural numbers, with player 1 winning in the event that the resulting sequence x is in A. We will give proofs of Gale and Stewart’s classic results that any open subset A of Baire space leads to the game G(A) being determined (i.e. one of the players has a winning strategy) and that the Axiom of Determinacy (stating that all games are determined) contradicts the Axiom of Choice. With the former we recreate Blackwell’s groundbreaking proof of a classical result about co-analytic sets. A family U of subsets of Baire space is said to have the reduction property if for any B and C in U, there are respective disjoint subsets B* of B and C* of C in U with the same union as B and C; Blackwell proves that the co-analytic sets have the reduction property. Blackwell’s new proof technique with this old result revitalized this area of descriptive set theory and began the development for a slew of new results.